{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Introduction to Probability and Statistics\n",
    "In this notebook, we will play around with some of the concepts we have previously discussed. Many concepts from probability and statistics are well-represented in major libraries for data processing in Python, such as `numpy` and `pandas`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import pandas as pd\n",
    "import random\n",
    "import matplotlib.pyplot as plt"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Random Variables and Distributions\n",
    "Let's start with drawing a sample of 30 values from a uniform distribution from 0 to 9. We will also compute mean and variance."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Sample: [4, 8, 5, 10, 5, 1, 1, 1, 7, 9, 7, 0, 2, 7, 3, 5, 9, 8, 3, 10, 2, 9, 2, 9, 9, 8, 1, 8, 7, 3]\n",
      "Mean = 5.433333333333334\n",
      "Variance = 10.178888888888887\n"
     ]
    }
   ],
   "source": [
    "sample = [ random.randint(0,10) for _ in range(30) ]\n",
    "print(f\"Sample: {sample}\")\n",
    "print(f\"Mean = {np.mean(sample)}\")\n",
    "print(f\"Variance = {np.var(sample)}\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "To visually estimate how many different values are there in the sample, we can plot the **histogram**:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.hist(sample)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Analyzing Real Data\n",
    "\n",
    "Mean and variance are very important when analyzing real-world data. Let's load the data about baseball players from [SOCR MLB Height/Weight Data](http://wiki.stat.ucla.edu/socr/index.php/SOCR_Data_MLB_HeightsWeights)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style scoped>\n",
       "    .dataframe tbody tr th:only-of-type {\n",
       "        vertical-align: middle;\n",
       "    }\n",
       "\n",
       "    .dataframe tbody tr th {\n",
       "        vertical-align: top;\n",
       "    }\n",
       "\n",
       "    .dataframe thead th {\n",
       "        text-align: right;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>Name</th>\n",
       "      <th>Team</th>\n",
       "      <th>Role</th>\n",
       "      <th>Height</th>\n",
       "      <th>Weight</th>\n",
       "      <th>Age</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>Adam_Donachie</td>\n",
       "      <td>BAL</td>\n",
       "      <td>Catcher</td>\n",
       "      <td>74</td>\n",
       "      <td>180.0</td>\n",
       "      <td>22.99</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>Paul_Bako</td>\n",
       "      <td>BAL</td>\n",
       "      <td>Catcher</td>\n",
       "      <td>74</td>\n",
       "      <td>215.0</td>\n",
       "      <td>34.69</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>Ramon_Hernandez</td>\n",
       "      <td>BAL</td>\n",
       "      <td>Catcher</td>\n",
       "      <td>72</td>\n",
       "      <td>210.0</td>\n",
       "      <td>30.78</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>Kevin_Millar</td>\n",
       "      <td>BAL</td>\n",
       "      <td>First_Baseman</td>\n",
       "      <td>72</td>\n",
       "      <td>210.0</td>\n",
       "      <td>35.43</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>Chris_Gomez</td>\n",
       "      <td>BAL</td>\n",
       "      <td>First_Baseman</td>\n",
       "      <td>73</td>\n",
       "      <td>188.0</td>\n",
       "      <td>35.71</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>...</th>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1029</th>\n",
       "      <td>Brad_Thompson</td>\n",
       "      <td>STL</td>\n",
       "      <td>Relief_Pitcher</td>\n",
       "      <td>73</td>\n",
       "      <td>190.0</td>\n",
       "      <td>25.08</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1030</th>\n",
       "      <td>Tyler_Johnson</td>\n",
       "      <td>STL</td>\n",
       "      <td>Relief_Pitcher</td>\n",
       "      <td>74</td>\n",
       "      <td>180.0</td>\n",
       "      <td>25.73</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1031</th>\n",
       "      <td>Chris_Narveson</td>\n",
       "      <td>STL</td>\n",
       "      <td>Relief_Pitcher</td>\n",
       "      <td>75</td>\n",
       "      <td>205.0</td>\n",
       "      <td>25.19</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1032</th>\n",
       "      <td>Randy_Keisler</td>\n",
       "      <td>STL</td>\n",
       "      <td>Relief_Pitcher</td>\n",
       "      <td>75</td>\n",
       "      <td>190.0</td>\n",
       "      <td>31.01</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1033</th>\n",
       "      <td>Josh_Kinney</td>\n",
       "      <td>STL</td>\n",
       "      <td>Relief_Pitcher</td>\n",
       "      <td>73</td>\n",
       "      <td>195.0</td>\n",
       "      <td>27.92</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "<p>1034 rows × 6 columns</p>\n",
       "</div>"
      ],
      "text/plain": [
       "                 Name Team            Role  Height  Weight    Age\n",
       "0       Adam_Donachie  BAL         Catcher      74   180.0  22.99\n",
       "1           Paul_Bako  BAL         Catcher      74   215.0  34.69\n",
       "2     Ramon_Hernandez  BAL         Catcher      72   210.0  30.78\n",
       "3        Kevin_Millar  BAL   First_Baseman      72   210.0  35.43\n",
       "4         Chris_Gomez  BAL   First_Baseman      73   188.0  35.71\n",
       "...               ...  ...             ...     ...     ...    ...\n",
       "1029    Brad_Thompson  STL  Relief_Pitcher      73   190.0  25.08\n",
       "1030    Tyler_Johnson  STL  Relief_Pitcher      74   180.0  25.73\n",
       "1031   Chris_Narveson  STL  Relief_Pitcher      75   205.0  25.19\n",
       "1032    Randy_Keisler  STL  Relief_Pitcher      75   190.0  31.01\n",
       "1033      Josh_Kinney  STL  Relief_Pitcher      73   195.0  27.92\n",
       "\n",
       "[1034 rows x 6 columns]"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df = pd.read_csv(\"../../data/SOCR_MLB.tsv\",sep='\\t', header=None, names=['Name','Team','Role','Height','Weight','Age'])\n",
    "df"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "> We are using a package called [**Pandas**](https://pandas.pydata.org/) here for data analysis. We will talk more about Pandas and working with data in Python later in this course.\n",
    "\n",
    "Let's compute average values for age, height and weight:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Age        28.736712\n",
       "Height     73.697292\n",
       "Weight    201.689255\n",
       "dtype: float64"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df[['Age','Height','Weight']].mean()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now let's focus on height, and compute standard deviation and variance: "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[74, 74, 72, 72, 73, 69, 69, 71, 76, 71, 73, 73, 74, 74, 69, 70, 72, 73, 75, 78]\n"
     ]
    }
   ],
   "source": [
    "print(list(df['Height'])[:20])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Mean = 73.6972920696325\n",
      "Variance = 5.316798081118074\n",
      "Standard Deviation = 2.3058183105175645\n"
     ]
    }
   ],
   "source": [
    "mean = df['Height'].mean()\n",
    "var = df['Height'].var()\n",
    "std = df['Height'].std()\n",
    "print(f\"Mean = {mean}\\nVariance = {var}\\nStandard Deviation = {std}\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In addition to mean, it makes sense to look at the median value and quartiles. They can be visualized using a **box plot**:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 720x144 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.figure(figsize=(10,2))\n",
    "plt.boxplot(df['Height'], vert=False, showmeans=True)\n",
    "plt.grid(color='gray', linestyle='dotted')\n",
    "plt.tight_layout()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can also make box plots of subsets of our dataset, for example, grouped by player role."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 720x576 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "df.boxplot(column='Height', by='Role', figsize=(10,8))\n",
    "plt.xticks(rotation='vertical')\n",
    "plt.tight_layout()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "> **Note**: This diagram suggests, that on average, the heights of first basemen are higher than heights of second basemen. Later we will learn how we can test this hypothesis more formally, and how to demonstrate that our data is statistically significant to show that.  \n",
    "\n",
    "Age, height and weight are all continuous random variables. What do you think their distribution is? A good way to find out is to plot the histogram of values: "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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Zq7/9GOAQ4GsjqE2SJEmr3DAv83Ym8GngJ5Jcm+Ql/abjuOfyCoCnAl9I8nngfcDLqurmYdUmSZIkzWSYV7F4wQzjJ04zdjZw9rBqkSRJkubLt5qWJEmSGgZkSZIkqWFAliRJkhoGZEmSJKlhQJYkSZIaBmRJkiSpYUCWJEmSGgZkSZIkqWFAliRJkhoGZEmSJKlhQJYkSZIaBmRJkiSpYUCWJEmSGgZkSZIkqWFAliRJkhoGZEmSJKlhQJYkSZIaBmRJkiSpYUCWJEmSGgZkSZIkqWFAliRJkhoGZEmSJKlhQJYkSZIaBmRJkiSpYUCWJEmSGgZkSZIkqWFAliRJkhoGZEmSJKlhQJYkSZIaBmRJkiSpYUCWJEmSGgZkSZIkqWFAliRJkhoGZEmSJKmx97AOnOR04Bjgxqo6vB97PfBS4Fv9bq+uqvP6bScBLwHuAn6rqj4yrNokrV5rN5076hJmteOUo0ddgiStesOcQX4H8Kxpxt9SVUf0H5Ph+FDgOOCw/jF/nWSvIdYmSZIkTWtoAbmqLgRunufuxwJbqurOqroa+Cpw1LBqkyRJkmaSqhrewZO1wIenLLE4EbgN2AZsrKpbkrwVuKiq3tXvdxrwj1X1vmmOuQHYADA2Nnbkli1bhlb/SrBr1y7WrFkz6jJWlFH3dPt1t47s3MM09iDYeceoqxi9dQftO7BjjfprdaWyr4NnT4fDvs5t/fr1F1fV+NTxoa1BnsHbgDcC1X8+FXgxkGn2nTa5V9VmYDPA+Ph4TUxMDKXQlWLr1q3Yo8EadU9PXOJraBdq47rdnLp9sb8lLT07jp8Y2LFG/bW6UtnXwbOnw2FfF25RfxpV1c7J20n+Fvhwf/da4OBm10cB1y9iadIPzPUiro3rdq/YkCpJkhb5Mm9JDmzu/jJwWX/7HOC4JPskeTRwCPDZxaxNkiRJguFe5u1MYAI4IMm1wOuAiSRH0C2f2AH8GkBVXZ7kLOCLwG7gFVV117BqkyRJkmYytIBcVS+YZvi0WfY/GTh5WPVIkiRJ8+E76UmSJEkNA7IkSZLUMCBLkiRJDQOyJEmS1DAgS5IkSQ0DsiRJktQwIEuSJEkNA7IkSZLUMCBLkiRJDQOyJEmS1DAgS5IkSQ0DsiRJktQwIEuSJEkNA7IkSZLUMCBLkiRJDQOyJEmS1DAgS5IkSQ0DsiRJktQwIEuSJEkNA7IkSZLUMCBLkiRJDQOyJEmS1DAgS5IkSQ0DsiRJktQwIEuSJEkNA7IkSZLUMCBLkiRJDQOyJEmS1DAgS5IkSQ0DsiRJktQwIEuSJEkNA7IkSZLUMCBLkiRJDQOyJEmS1DAgS5IkSY2hBeQkpye5McllzdifJflSki8k+UCS/frxtUnuSHJp//H2YdUlSZIkzWaYM8jvAJ41Zex84PCq+n+ArwAnNduuqqoj+o+XDbEuSZIkaUZDC8hVdSFw85Sxj1bV7v7uRcCjhnV+SZIkaSFSVcM7eLIW+HBVHT7Ntn8A3ltV7+r3u5xuVvk24A+q6hMzHHMDsAFgbGzsyC1btgyp+pVh165drFmzZtRlLCvbr7t11u1jD4KddyxSMauIfe2sO2jfgR3L///DYV8Hz54Oh32d2/r16y+uqvGp43uPopgkrwF2A+/uh24AfrSqbkpyJPDBJIdV1W1TH1tVm4HNAOPj4zUxMbFIVS9PW7duxR7tmRM3nTvr9o3rdnPq9pH811nR7Gtnx/ETAzuW//+Hw74Onj0dDvu6cIt+FYskJwDHAMdXP31dVXdW1U397YuBq4DHLXZtkiRJ0qIG5CTPAn4f+KWq+l4z/vAke/W3HwMcAnxtMWuTJEmSYIhLLJKcCUwAByS5Fngd3VUr9gHOTwJwUX/FiqcCb0iyG7gLeFlV3TztgSVJkqQhGlpArqoXTDN82gz7ng2cPaxaJEmSpPnynfQkSZKkhgFZkiRJahiQJUmSpIYBWZIkSWoYkCVJkqSGAVmSJElqGJAlSZKkhgFZkiRJahiQJUmSpIYBWZIkSWoYkCVJkqSGAVmSJElqGJAlSZKkhgFZkiRJahiQJUmSpIYBWZIkSWoYkCVJkqSGAVmSJElqGJAlSZKkhgFZkiRJahiQJUmSpMa8AnKSJ89nTJIkSVru5juD/D/nOSZJkiQta3vPtjHJE4EnAQ9P8qpm00OBvYZZmCRJkjQKswZk4AHAmn6/hzTjtwHPHVZRkiRJ0qjMGpCr6gLggiTvqKprFqkmSZIkaWTmmkGetE+SzcDa9jFV9bRhFCVJkiSNynwD8v8C3g78HXDX8MqRJEmSRmu+AXl3Vb1tqJVIkiRJS8B8L/P2D0l+PcmBSR42+THUyiRJkqQRmO8M8gn9599rxgp4zGDLkSRJkkZrXgG5qh497EIkSZKkpWBeATnJi6Ybr6p3DrYcSZIkabTmu8TiCc3tBwJPBy4BDMiSJElaUea7xOI32/tJ9gX+frbHJDkdOAa4saoO78ceBryX7nrKO4DnV9Ut/baTgJfQXUbut6rqI3vyRCRJkqRBmO8M8lTfAw6ZY593AG/lnrPMm4CPVdUpSTb1938/yaHAccBhwCOBf07yuKrymsuSVpW1m84d2LE2rtvNiQM83o5Tjh7YsSRpKZvvGuR/oLtqBcBewOOBs2Z7TFVdmGTtlOFjgYn+9hnAVuD3+/EtVXUncHWSrwJHAZ+eT32SJEnSoKSq5t4p+fnm7m7gmqq6dh6PWwt8uFli8Z2q2q/ZfktV7Z/krcBFVfWufvw04B+r6n3THHMDsAFgbGzsyC1btsxZ/2q2a9cu1qxZM+oylpXt19066/axB8HOOxapmFXEvg7eoHu67qB9B3ewZczvq4NnT4fDvs5t/fr1F1fV+NTx+a5BviDJGHe/WO/KQRYHZLrTzlDLZmAzwPj4eE1MTAy4lJVl69at2KM9M9efpDeu282p2xe6Okkzsa+DN+ie7jh+YmDHWs78vjp49nQ47OvCzeud9JI8H/gs8Dzg+cBnkjx3AefbmeTA/pgHAjf249cCBzf7PQq4fgHHlyRJku6T+b7V9GuAJ1TVCVX1Irr1wX+4gPOdw93vyncC8KFm/Lgk+yR5NN0LAD+7gONLkiRJ98l8//Z2v6q6sbl/E3OE6yRn0r0g74Ak1wKvA04BzkryEuDrdDPSVNXlSc4Cvki3xvkVXsFCkiRJozDfgPxPST4CnNnf/xXgvNkeUFUvmGHT02fY/2Tg5HnWI0mSJA3FrAE5yY8DY1X1e0n+M/AUuhfUfRp49yLUJ0mSJC2qudYg/wVwO0BVvb+qXlVVv0M3e/wXwy1NkiRJWnxzBeS1VfWFqYNVtY3u7aIlSZKkFWWugPzAWbY9aJCFSJIkSUvBXAH5c0leOnWwvwrFxcMpSZIkSRqdua5i8UrgA0mO5+5APA48APjlIdYlSZIkjcSsAbmqdgJPSrIeOLwfPreq/mXolUmSJEkjMK/rIFfVx4GPD7kWSZIkaeTm+1bTkiRJ0qpgQJYkSZIaBmRJkiSpYUCWJEmSGgZkSZIkqWFAliRJkhoGZEmSJKlhQJYkSZIaBmRJkiSpYUCWJEmSGgZkSZIkqWFAliRJkhoGZEmSJKlhQJYkSZIaBmRJkiSpYUCWJEmSGgZkSZIkqWFAliRJkhoGZEmSJKlhQJYkSZIaBmRJkiSpYUCWJEmSGgZkSZIkqWFAliRJkhoGZEmSJKlhQJYkSZIaBmRJkiSpsfdinzDJTwDvbYYeA7wW2A94KfCtfvzVVXXe4lYnSZKk1W7RA3JVfRk4AiDJXsB1wAeA/wa8par+fLFrkiRJkiaNeonF04GrquqaEdchSZIkAZCqGt3Jk9OBS6rqrUleD5wI3AZsAzZW1S3TPGYDsAFgbGzsyC1btixewcvQrl27WLNmzajLWFa2X3frrNvHHgQ771ikYlYR+zp4g+7puoP2HdzBljG/rw6ePR0O+zq39evXX1xV41PHRxaQkzwAuB44rKp2JhkDvg0U8EbgwKp68WzHGB8fr23btg2/2GVs69atTExMjLqMZWXtpnNn3b5x3W5O3b7oq5NWPPs6eKutpztOOXpRzuP31cGzp8NhX+eWZNqAPMolFr9IN3u8E6CqdlbVXVX1feBvgaNGWJskSZJWqVFOLbwAOHPyTpIDq+qG/u4vA5eNpCoN3VwztJIkSaM0koCc5IeA/wj8WjP8piRH0C2x2DFlmyRJkrQoRhKQq+p7wA9PGXvhKGqRJEmSWqO+zJskSZK0pBiQJUmSpIYBWZIkSWoYkCVJkqSGAVmSJElqGJAlSZKkhgFZkiRJahiQJUmSpIYBWZIkSWoYkCVJkqSGAVmSJElqGJAlSZKkhgFZkiRJahiQJUmSpIYBWZIkSWoYkCVJkqSGAVmSJElqGJAlSZKkhgFZkiRJahiQJUmSpIYBWZIkSWoYkCVJkqSGAVmSJElqGJAlSZKkhgFZkiRJahiQJUmSpIYBWZIkSWoYkCVJkqSGAVmSJElqGJAlSZKkhgFZkiRJahiQJUmSpIYBWZIkSWoYkCVJkqTG3qM4aZIdwO3AXcDuqhpP8jDgvcBaYAfw/Kq6ZRT1SZIkafUa5Qzy+qo6oqrG+/ubgI9V1SHAx/r7kiRJ0qJaSkssjgXO6G+fATxndKVIkiRptUpVLf5Jk6uBW4AC/qaqNif5TlXt1+xzS1XtP81jNwAbAMbGxo7csmXLIlW9PO3atYs1a9aMuox72H7draMu4T4ZexDsvGPUVaw89nXwVltP1x2076KcZyl+X13u7Olw2Ne5rV+//uJmNcMPjGQNMvDkqro+ySOA85N8ab4PrKrNwGaA8fHxmpiYGFKJK8PWrVtZaj06cdO5oy7hPtm4bjenbh/Vf52Vy74O3mrr6Y7jJxblPEvx++pyZ0+Hw74u3EiWWFTV9f3nG4EPAEcBO5McCNB/vnEUtUmSJGl1W/SAnOTBSR4yeRt4BnAZcA5wQr/bCcCHFrs2SZIkaRR/exsDPpBk8vzvqap/SvI54KwkLwG+DjxvBLVJkiRplVv0gFxVXwN+aprxm4CnL3Y9kiRJUmspXeZNkiRJGjkDsiRJktQwIEuSJEkNA7IkSZLUMCBLkiRJDQOyJEmS1DAgS5IkSQ0DsiRJktQwIEuSJEkNA7IkSZLUMCBLkiRJjb1HXYAkSYOwdtO5i3Kejet2c+ICzrXjlKOHUI2kYXAGWZIkSWoYkCVJkqSGAVmSJElqGJAlSZKkhgFZkiRJahiQJUmSpIYBWZIkSWoYkCVJkqSGAVmSJElqGJAlSZKkhgFZkiRJahiQJUmSpIYBWZIkSWoYkCVJkqSGAVmSJElqGJAlSZKkhgFZkiRJauw96gI0eGs3nfuD2xvX7ebE5r4kSZJm5wyyJEmS1DAgS5IkSQ0DsiRJktQwIEuSJEmNRQ/ISQ5O8vEkVyS5PMlv9+OvT3Jdkkv7j2cvdm2SJEnSKK5isRvYWFWXJHkIcHGS8/ttb6mqPx9BTZIkSRIwgoBcVTcAN/S3b09yBXDQYtchSZIkTSdVNbqTJ2uBC4HDgVcBJwK3AdvoZplvmeYxG4ANAGNjY0du2bJlscpdNrZfd+sPbo89CHbeMcJiViB7Ohz2dfDs6XAstK/rDtp38MWsELt27WLNmjWjLmPFsa9zW79+/cVVNT51fGQBOcka4ALg5Kp6f5Ix4NtAAW8EDqyqF892jPHx8dq2bdvwi11mpr5RyKnbfT+YQbKnw2FfB8+eDsdC+7rjlKOHUM3KsHXrViYmJkZdxopjX+eWZNqAPJKrWCS5P3A28O6qej9AVe2sqruq6vvA3wJHjaI2SZIkrW6juIpFgNOAK6rqzc34gc1uvwxctti1SZIkSaP429uTgRcC25Nc2o+9GnhBkiPolljsAH5tBLVJkjQU7fK3pcglINLdRnEVi08CmWbTeYtdiyRJkjSV76QnSZIkNQzIkiRJUsOALEmSJDUMyJIkSVLDgCxJkiQ1DMiSJElSw4AsSZIkNQzIkiRJUsOALEmSJDUMyJIkSVLDgCxJkiQ1DMiSJElSw4AsSZIkNQzIkiRJUsOALEmSJDUMyJIkSVLDgCxJkiQ1DMiSJElSw4AsSZIkNQzIkiRJUsOALEmSJDUMyJIkSVLDgCxJkiQ1DMiSJElSY+9RF7Acrd107qhLkCRJ0pA4gyxJkiQ1nEGWJEkj/evoxnW7OXGO8+845ehFqkZyBlmSJEm6BwOyJEmS1DAgS5IkSQ0DsiRJktQwIEuSJEkNA7IkSZLUMCBLkiRJDa+DLEmSdB8txXfZba8v7XWk98ySm0FO8qwkX07y1SSbRl2PJEmSVpclNYOcZC/gr4D/CFwLfC7JOVX1xdFWJkmSRmkpztAuJ0u9f0tthnupzSAfBXy1qr5WVf8GbAGOHXFNkiRJWkVSVaOu4QeSPBd4VlX9an//hcDPVNVvNPtsADb0d38C+PKiF7q8HAB8e9RFrDD2dDjs6+DZ0+Gwr4NnT4fDvs7tx6rq4VMHl9QSCyDTjN0jwVfVZmDz4pSz/CXZVlXjo65jJbGnw2FfB8+eDod9HTx7Ohz2deGW2hKLa4GDm/uPAq4fUS2SJElahZZaQP4ccEiSRyd5AHAccM6Ia5IkSdIqsqSWWFTV7iS/AXwE2As4vaouH3FZy53LUQbPng6HfR08ezoc9nXw7Olw2NcFWlIv0pMkSZJGbaktsZAkSZJGyoAsSZIkNQzIy1yS05PcmOSyKeO/2b9l9+VJ3tSMn9S/jfeXkzxz8Ste+qbraZIjklyU5NIk25Ic1Wyzp3NIcnCSjye5ov+a/O1+/GFJzk9yZf95/+Yx9nUOs/T1z5J8KckXknwgyX7NY+zrLGbqabP9d5NUkgOaMXs6h9n66s+rhZnl/78/rwahqvxYxh/AU4GfBi5rxtYD/wzs099/RP/5UODzwD7Ao4GrgL1G/RyW2scMPf0o8Iv97WcDW+3pHvX0QOCn+9sPAb7S9+5NwKZ+fBPwp/Z1IH19BrB3P/6n9vW+97S/fzDdi8ivAQ6wp/e9r/68GkpP/Xk1gA9nkJe5qroQuHnK8MuBU6rqzn6fG/vxY4EtVXVnVV0NfJXu7b3VmKGnBTy0v70vd1+f257OQ1XdUFWX9LdvB64ADqLr3xn9bmcAz+lv29d5mKmvVfXRqtrd73YR3TXlwb7OaZavVYC3AP8f93wDK3s6D7P01Z9XCzRLT/15NQAG5JXpccDPJflMkguSPKEfPwj4RrPftdz9jV+zeyXwZ0m+Afw5cFI/bk/3UJK1wH8APgOMVdUN0H2zBx7R72Zf99CUvrZeDPxjf9u+7oG2p0l+Cbiuqj4/ZTd7uoemfK3682oApvT0lfjz6j4zIK9MewP7Az8L/B5wVpIwj7fy1oxeDvxOVR0M/A5wWj9uT/dAkjXA2cArq+q22XadZsy+zmCmviZ5DbAbePfk0DQPt6/TaHtK18PXAK+dbtdpxuzpDKb5WvXn1X00TU/9eTUABuSV6Vrg/dX5LPB94AB8K+/74gTg/f3t/8Xdf5ayp/OU5P5038TfXVWTvdyZ5MB++4HA5J9X7es8zdBXkpwAHAMcX/0CROzrvEzT08fSrdn8fJIddH27JMmPYE/nbYavVX9e3Qcz9NSfVwNgQF6ZPgg8DSDJ44AHAN+me9vu45Lsk+TRwCHAZ0dV5DJzPfDz/e2nAVf2t+3pPPQzQqcBV1TVm5tN59B9M6f//KFm3L7OYaa+JnkW8PvAL1XV95qH2Nc5TNfTqtpeVY+oqrVVtZYuaPx0VX0Tezovs3wP+CD+vFqQWXrqz6sBWFJvNa09l+RMYAI4IMm1wOuA04HT012m7N+AE/oZpMuTnAV8ke5Phq+oqrtGU/nSNUNPXwr8jyR7A/8KbACoKns6P08GXghsT3JpP/Zq4BS6P6m+BPg68Dywr3tgpr7+Jd0r1c/vfoZyUVW9zL7Oy7Q9rarzptvZns7bTF+r/rxauJl66s+rAfCtpiVJkqSGSywkSZKkhgFZkiRJahiQJUmSpIYBWZIkSWoYkCVJkqSGAVmSlqAkb0nyyub+R5L8XXP/1CSvmuGxb0jyC3Mc//VJfnea8f2S/Pp9KF2Slj0DsiQtTf8beBJAkvvRvbvYYc32JwGfmu6BVfXaqvrnBZ53P8CALGlVMyBL0tL0KfqATBeMLwNuT7J/kn2AxwMkuSDJxf0M8+Tbdr8jyXP7289O8qUkn0zyl0k+3Jzj0CRbk3wtyW/1Y6cAj01yaZI/W4wnKklLje+kJ0lLUFVdn2R3kh+lC8qfBg4CngjcClwBvAU4tqq+leRXgJOBF08eI8kDgb8BnlpVV/fvEtn6SWA98BDgy0neBmwCDq+qI4b6BCVpCTMgS9LSNTmL/CTgzXQB+Ul0Afk64Bnc/XbSewE3THn8TwJfq6qr+/tn0r/tbO/cqroTuDPJjcDYkJ6HJC0rBmRJWrom1yGvo1ti8Q1gI3Ab8C/AQVX1xFkenzmOf2dz+y78mSBJgGuQJWkp+xRwDHBzVd1VVTfTvYjuicB7gYcneSJAkvsnOWzK478EPCbJ2v7+r8zjnLfTLbmQpFXLgCxJS9d2uqtXXDRl7NaquhF4LvCnST4PXMrdL+oDoKruoLsixT8l+SSwk255xoyq6ibgU0ku80V6klarVNWoa5AkDUmSNVW1K91C5b8Crqyqt4y6LklaypxBlqSV7aVJLgUuB/alu6qFJGkWziBLkiRJDWeQJUmSpIYBWZIkSWoYkCVJkqSGAVmSJElqGJAlSZKkxv8FiHh2DxCDPowAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<Figure size 720x432 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "df['Weight'].hist(bins=15, figsize=(10,6))\n",
    "plt.suptitle('Weight distribution of MLB Players')\n",
    "plt.xlabel('Weight')\n",
    "plt.ylabel('Count')\n",
    "plt.tight_layout()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Normal Distribution\n",
    "\n",
    "Let's create an artificial sample of weights that follows a normal distribution with the same mean and variance as our real data:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([73.46072234, 70.40678311, 70.23689776, 73.81190675, 72.41091792,\n",
       "       76.00127651, 71.91641414, 77.18162239, 76.7173353 , 73.93996587,\n",
       "       74.2862748 , 76.88034696, 72.15184905, 74.43537605, 76.37723417,\n",
       "       65.66976051, 74.3200533 , 77.3235274 , 72.8840488 , 77.50300255])"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "generated = np.random.normal(mean, std, 1000)\n",
    "generated[:20]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 720x432 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.figure(figsize=(10,6))\n",
    "plt.hist(generated, bins=15)\n",
    "plt.tight_layout()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 720x432 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.figure(figsize=(10,6))\n",
    "plt.hist(np.random.normal(0,1,50000), bins=300)\n",
    "plt.tight_layout()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Since most values in real life are normally distributed, we should not use a uniform random number generator to generate sample data. Here is what happens if we try to generate weights with a uniform distribution (generated by `np.random.rand`):"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 720x432 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "wrong_sample = np.random.rand(1000)*2*std+mean-std\n",
    "plt.figure(figsize=(10,6))\n",
    "plt.hist(wrong_sample)\n",
    "plt.tight_layout()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Confidence Intervals\n",
    "\n",
    "Let's now calculate confidence intervals for the weights and heights of baseball players. We will use the code [from this stackoverflow discussion](https://stackoverflow.com/questions/15033511/compute-a-confidence-interval-from-sample-data):"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "p=0.85, mean = 201.73 ± 0.94\n",
      "p=0.90, mean = 201.73 ± 1.08\n",
      "p=0.95, mean = 201.73 ± 1.28\n"
     ]
    }
   ],
   "source": [
    "import scipy.stats\n",
    "\n",
    "def mean_confidence_interval(data, confidence=0.95):\n",
    "    a = 1.0 * np.array(data)\n",
    "    n = len(a)\n",
    "    m, se = np.mean(a), scipy.stats.sem(a)\n",
    "    h = se * scipy.stats.t.ppf((1 + confidence) / 2., n-1)\n",
    "    return m, h\n",
    "\n",
    "for p in [0.85, 0.9, 0.95]:\n",
    "    m, h = mean_confidence_interval(df['Weight'].fillna(method='pad'),p)\n",
    "    print(f\"p={p:.2f}, mean = {m:.2f} ± {h:.2f}\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Hypothesis Testing\n",
    "\n",
    "Let's explore different roles in our baseball players dataset:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style scoped>\n",
       "    .dataframe tbody tr th:only-of-type {\n",
       "        vertical-align: middle;\n",
       "    }\n",
       "\n",
       "    .dataframe tbody tr th {\n",
       "        vertical-align: top;\n",
       "    }\n",
       "\n",
       "    .dataframe thead th {\n",
       "        text-align: right;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>Height</th>\n",
       "      <th>Weight</th>\n",
       "      <th>Count</th>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Role</th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>Catcher</th>\n",
       "      <td>72.723684</td>\n",
       "      <td>204.328947</td>\n",
       "      <td>76</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Designated_Hitter</th>\n",
       "      <td>74.222222</td>\n",
       "      <td>220.888889</td>\n",
       "      <td>18</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>First_Baseman</th>\n",
       "      <td>74.000000</td>\n",
       "      <td>213.109091</td>\n",
       "      <td>55</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Outfielder</th>\n",
       "      <td>73.010309</td>\n",
       "      <td>199.113402</td>\n",
       "      <td>194</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Relief_Pitcher</th>\n",
       "      <td>74.374603</td>\n",
       "      <td>203.517460</td>\n",
       "      <td>315</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Second_Baseman</th>\n",
       "      <td>71.362069</td>\n",
       "      <td>184.344828</td>\n",
       "      <td>58</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Shortstop</th>\n",
       "      <td>71.903846</td>\n",
       "      <td>182.923077</td>\n",
       "      <td>52</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Starting_Pitcher</th>\n",
       "      <td>74.719457</td>\n",
       "      <td>205.163636</td>\n",
       "      <td>221</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Third_Baseman</th>\n",
       "      <td>73.044444</td>\n",
       "      <td>200.955556</td>\n",
       "      <td>45</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "                      Height      Weight  Count\n",
       "Role                                           \n",
       "Catcher            72.723684  204.328947     76\n",
       "Designated_Hitter  74.222222  220.888889     18\n",
       "First_Baseman      74.000000  213.109091     55\n",
       "Outfielder         73.010309  199.113402    194\n",
       "Relief_Pitcher     74.374603  203.517460    315\n",
       "Second_Baseman     71.362069  184.344828     58\n",
       "Shortstop          71.903846  182.923077     52\n",
       "Starting_Pitcher   74.719457  205.163636    221\n",
       "Third_Baseman      73.044444  200.955556     45"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df.groupby('Role').agg({ 'Height' : 'mean', 'Weight' : 'mean', 'Age' : 'count'}).rename(columns={ 'Age' : 'Count'})"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's test the hypothesis that First Basemen are taller than Second Basemen. The simplest way to do this is to test the confidence intervals:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Conf=0.85, 1st basemen height: 73.62..74.38, 2nd basemen height: 71.04..71.69\n",
      "Conf=0.90, 1st basemen height: 73.56..74.44, 2nd basemen height: 70.99..71.73\n",
      "Conf=0.95, 1st basemen height: 73.47..74.53, 2nd basemen height: 70.92..71.81\n"
     ]
    }
   ],
   "source": [
    "for p in [0.85,0.9,0.95]:\n",
    "    m1, h1 = mean_confidence_interval(df.loc[df['Role']=='First_Baseman',['Height']],p)\n",
    "    m2, h2 = mean_confidence_interval(df.loc[df['Role']=='Second_Baseman',['Height']],p)\n",
    "    print(f'Conf={p:.2f}, 1st basemen height: {m1-h1[0]:.2f}..{m1+h1[0]:.2f}, 2nd basemen height: {m2-h2[0]:.2f}..{m2+h2[0]:.2f}')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can see that the intervals do not overlap.\n",
    "\n",
    "A statistically more correct way to prove the hypothesis is to use a **Student t-test**:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "T-value = 7.65\n",
      "P-value: 9.137321189738925e-12\n"
     ]
    }
   ],
   "source": [
    "from scipy.stats import ttest_ind\n",
    "\n",
    "tval, pval = ttest_ind(df.loc[df['Role']=='First_Baseman',['Height']], df.loc[df['Role']=='Second_Baseman',['Height']],equal_var=False)\n",
    "print(f\"T-value = {tval[0]:.2f}\\nP-value: {pval[0]}\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The two values returned by the `ttest_ind` function are:\n",
    "* p-value can be considered as the probability of two distributions having the same mean. In our case, it is very low, meaning that there is strong evidence supporting that first basemen are taller.\n",
    "* t-value is the intermediate value of normalized mean difference that is used in the t-test, and it is compared against a threshold value for a given confidence value."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Simulating a Normal Distribution with the Central Limit Theorem\n",
    "\n",
    "The pseudo-random generator in Python is designed to give us a uniform distribution. If we want to create a generator for normal distribution, we can use the central limit theorem. To get a normally distributed value we will just compute a mean of a uniform-generated sample."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAsgAAAGoCAYAAABbtxOxAAAAOXRFWHRTb2Z0d2FyZQBNYXRwbG90bGliIHZlcnNpb24zLjUuMiwgaHR0cHM6Ly9tYXRwbG90bGliLm9yZy8qNh9FAAAACXBIWXMAAAsTAAALEwEAmpwYAAARLElEQVR4nO3df4zkd13H8ddblgbkR4DcghU4Fgghlj/4kbOIGFNDMEiNQIIJJGI1mFMjBJREL/yh/FnjryZGMRWQGn6FQPkRriqkkqCJEq9QQpuCIFQsXLg2KKAxIS0f/9g5eLfdc7fznd3v7O3jkUxu5rszO+/93Ox+n/e9mZ0aYwQAANj2A3MPAAAA60QgAwBAI5ABAKARyAAA0AhkAABoNg7yzo4dOza2trYO8i4BAGBHN910011jjM37bj/QQN7a2sqZM2cO8i4BAGBHVfXvO233FAsAAGgEMgAANAIZAAAagQwAAI1ABgCARiADAEAjkAEAoBHIAADQCGQAAGgEMgAANAIZAAAagQwAAI1ABgCARiADAEAjkAEAoBHIAADQCGQAAGg25h4AgAdm69TpuUeYxe1XXzn3CMAR4QgyAAA0AhkAABqBDAAAjUAGAIBGIAMAQCOQAQCgEcgAANAIZAAAaAQyAAA0AhkAABqBDAAAjUAGAIBGIAMAQCOQAQCgEcgAANAIZAAAaAQyAAA0AhkAABqBDAAAjUAGAIBGIAMAQCOQAQCgEcgAANAIZAAAaAQyAAA0AhkAABqBDAAAjUAGAIBGIAMAQCOQAQCg2TWQq+qJVfXxqrqtqm6tqtcttj+mqj5WVV9Y/Pno/R8XAAD2116OIN+d5A1jjB9J8mNJfqOqLktyKsmNY4ynJblxcRkAAA61XQN5jHF2jPGpxflvJ7ktyeOTvCTJdYurXZfkpfs0IwAAHJgH9BzkqtpK8uwkn0zyuDHG2WQ7opM89gK3OVlVZ6rqzJ133jlxXAAA2F97DuSqeniS9yd5/RjjW3u93Rjj2jHGiTHGic3NzWVmBACAA7OnQK6qB2c7jt85xrh+sfnrVXXp4uOXJjm3PyMCAMDB2ctvsagkb01y2xjjj9uHPpzkqsX5q5J8aPXjAQDAwdrYw3Wen+RVST5bVTcvtr0xydVJ3ltVr07ylSQ/vy8TAgDAAdo1kMcY/5ikLvDhF6x2HAAAmJd30gMAgEYgAwBAI5ABAKARyAAA0AhkAABoBDIAADQCGQAAGoEMAACNQAYAgEYgAwBAI5ABAKARyAAA0AhkAABoBDIAADQCGQAAGoEMAACNQAYAgEYgAwBAI5ABAKARyAAA0AhkAABoBDIAADQCGQAAGoEMAACNQAYAgEYgAwBAI5ABAKARyAAA0AhkAABoBDIAADQCGQAAGoEMAACNQAYAgEYgAwBAI5ABAKARyAAA0AhkAABoBDIAADQCGQAAGoEMAACNQAYAgEYgAwBAI5ABAKARyAAA0AhkAABoNuYeAGCKrVOn5x4BgIuMI8gAANAIZAAAaAQyAAA0AhkAABqBDAAAjUAGAIBGIAMAQCOQAQCgEcgAANAIZAAAaAQyAAA0AhkAABqBDAAAjUAGAIBGIAMAQCOQAQCgEcgAANAIZAAAaAQyAAA0AhkAABqBDAAAjUAGAIBGIAMAQCOQAQCgEcgAANAIZAAAaAQyAAA0AhkAABqBDAAAjUAGAIBm10CuqrdV1bmquqVte1NVfbWqbl6cXry/YwIAwMHYyxHktyd50Q7b/2SM8azF6YbVjgUAAPPYNZDHGJ9I8o0DmAUAAGa3MeG2r6mqX0xyJskbxhj/udOVqupkkpNJcvz48Ql3BwBHz9ap03OPcOBuv/rKuUfgiFv2RXpvTvLUJM9KcjbJH13oimOMa8cYJ8YYJzY3N5e8OwAAOBhLBfIY4+tjjHvGGN9N8pdJLl/tWAAAMI+lArmqLm0XX5bklgtdFwAADpNdn4NcVe9OckWSY1V1R5LfS3JFVT0ryUhye5Jf3b8RAQDg4OwayGOMV+6w+a37MAsAAMzOO+kBAEAjkAEAoBHIAADQCGQAAGgEMgAANAIZAAAagQwAAI1ABgCARiADAEAjkAEAoBHIAADQbMw9AADsxdap03OPABwRjiADAEAjkAEAoBHIAADQCGQAAGgEMgAANAIZAAAagQwAAI1ABgCARiADAEAjkAEAoBHIAADQCGQAAGgEMgAANAIZAAAagQwAAI1ABgCARiADAEAjkAEAoBHIAADQCGQAAGgEMgAANAIZAAAagQwAAI1ABgCARiADAEAjkAEAoBHIAADQCGQAAGgEMgAANAIZAAAagQwAAI1ABgCARiADAECzMfcAwGpsnTo99wgAcFFwBBkAABqBDAAAjUAGAIBGIAMAQCOQAQCgEcgAANAIZAAAaAQyAAA0AhkAABqBDAAAjUAGAIBGIAMAQCOQAQCgEcgAANAIZAAAaAQyAAA0AhkAABqBDAAAjUAGAIBGIAMAQCOQAQCgEcgAANAIZAAAaAQyAAA0AhkAABqBDAAAjUAGAIBGIAMAQCOQAQCgEcgAANDsGshV9baqOldVt7Rtj6mqj1XVFxZ/Pnp/xwQAgIOxlyPIb0/yovtsO5XkxjHG05LcuLgMAACH3q6BPMb4RJJv3GfzS5Jctzh/XZKXrnYsAACYx8aSt3vcGONskowxzlbVYy90xao6meRkkhw/fnzJuwMAjoqtU6fnHmEWt1995dwjsLDvL9IbY1w7xjgxxjixubm533cHAACTLBvIX6+qS5Nk8ee51Y0EAADzWTaQP5zkqsX5q5J8aDXjAADAvPbya97eneSfkjy9qu6oqlcnuTrJC6vqC0leuLgMAACH3q4v0htjvPICH3rBimcBAIDZeSc9AABoBDIAADQCGQAAGoEMAACNQAYAgEYgAwBAI5ABAKARyAAA0AhkAABoBDIAADQCGQAAGoEMAACNQAYAgEYgAwBAI5ABAKARyAAA0AhkAABoBDIAADQCGQAAGoEMAACNQAYAgEYgAwBAI5ABAKARyAAA0AhkAABoBDIAADQCGQAAGoEMAACNQAYAgEYgAwBAI5ABAKARyAAA0AhkAABoBDIAADQCGQAAGoEMAACNQAYAgEYgAwBAI5ABAKARyAAA0AhkAABoBDIAADQCGQAAGoEMAACNQAYAgEYgAwBAI5ABAKARyAAA0AhkAABoBDIAADQCGQAAGoEMAACNQAYAgEYgAwBAI5ABAKARyAAA0AhkAABoBDIAADQCGQAAGoEMAACNQAYAgEYgAwBAI5ABAKARyAAA0AhkAABoBDIAADQCGQAAGoEMAACNQAYAgEYgAwBAI5ABAKARyAAA0AhkAABoBDIAADQCGQAAGoEMAADNxpQbV9XtSb6d5J4kd48xTqxiKAAAmMukQF74qTHGXSv4PAAAMDtPsQAAgGZqII8kH62qm6rq5CoGAgCAOU19isXzxxhfq6rHJvlYVX1ujPGJfoVFOJ9MkuPHj0+8OwCAi9PWqdNzjzCL26++cu4R7mfSEeQxxtcWf55L8oEkl+9wnWvHGCfGGCc2Nzen3B0AAOy7pQO5qh5WVY84fz7JTye5ZVWDAQDAHKY8xeJxST5QVec/z7vGGH+7kqkAAGAmSwfyGONLSZ65wlkAAGB2fs0bAAA0AhkAABqBDAAAjUAGAIBGIAMAQCOQAQCgEcgAANAIZAAAaAQyAAA0AhkAABqBDAAAjUAGAIBGIAMAQCOQAQCgEcgAANAIZAAAaAQyAAA0AhkAABqBDAAAjUAGAIBGIAMAQCOQAQCgEcgAANAIZAAAaAQyAAA0AhkAABqBDAAAjUAGAIBGIAMAQLMx9wCwalunTs89AgBwiDmCDAAAjUAGAIBGIAMAQCOQAQCgEcgAANAIZAAAaAQyAAA0AhkAABqBDAAAjUAGAIBGIAMAQCOQAQCgEcgAANAIZAAAaAQyAAA0AhkAABqBDAAAjUAGAIBGIAMAQCOQAQCgEcgAANAIZAAAaAQyAAA0AhkAABqBDAAAjUAGAIBGIAMAQCOQAQCgEcgAANAIZAAAaAQyAAA0AhkAABqBDAAAzcbcAxyUrVOn5x4BAIBDwBFkAABoBDIAADQCGQAAGoEMAACNQAYAgEYgAwBAI5ABAKARyAAA0AhkAABoBDIAADQCGQAAGoEMAACNQAYAgGZSIFfVi6rq81X1xao6taqhAABgLksHclU9KMmfJfmZJJcleWVVXbaqwQAAYA5TjiBfnuSLY4wvjTG+k+Q9SV6ymrEAAGAeGxNu+/gk/9Eu35Hkufe9UlWdTHJycfG/q+rzE+7zMDuW5K65h7gIWMfVsZarYy1XwzqujrVcHWu5Ghdcx/r9A57k3p6008YpgVw7bBv32zDGtUmunXA/F4WqOjPGODH3HIeddVwda7k61nI1rOPqWMvVsZarcdjWccpTLO5I8sR2+QlJvjZtHAAAmNeUQP6XJE+rqidX1SVJXpHkw6sZCwAA5rH0UyzGGHdX1WuS/F2SByV52xjj1pVNdvE58k8zWRHruDrWcnWs5WpYx9WxlqtjLVfjUK1jjXG/pw0DAMCR5Z30AACgEcgAANAI5In2+nbbVfWjVXVPVb18cfmJVfXxqrqtqm6tqtcd3NTradm1bNsfVFWfrqqP7P+062vKOlbVo6rqfVX1ucVj83kHM/V6mriWv7n43r6lqt5dVQ85mKnX025rWVVXVNU3q+rmxel393rbo2TZdbTPub8pj8nFx+1zFiZ+f6/nfmeM4bTkKdsvTvy3JE9JckmSzyS57ALX+/skNyR5+WLbpUmeszj/iCT/utNtj8ppylq2j/1Wkncl+cjcX89hXcck1yX5lcX5S5I8au6v6TCuZbbfSOnLSR66uPzeJL8099e0zmuZ5Iqdvnf3+vdwFE4T19E+Z0Vr2T5+5Pc5q1jLdd3vOII8zV7fbvu1Sd6f5Nz5DWOMs2OMTy3OfzvJbdneqR5VS69lklTVE5JcmeQt+z3omlt6HavqkUl+Mslbk2SM8Z0xxn/t+8Tra9JjMtu/JeihVbWR5AdztH9P/F7XctW3vdgsvRb2Ofcz6XFln3MvS6/lOu93BPI0O73d9r1+4FTV45O8LMlfXOiTVNVWkmcn+eTqRzw0pq7lNUl+O8l392m+w2LKOj4lyZ1J/mrx34ZvqaqH7eewa27ptRxjfDXJHyb5SpKzSb45xvjovk673nZdy4XnVdVnqupvquoZD/C2R8GUdfwe+5wk09fymtjnnDdlLdd2vyOQp9nL221fk+R3xhj37PgJqh6e7aNPrx9jfGu14x0qS69lVf1sknNjjJv2abbDZMpjciPJc5K8eYzx7CT/k+QoP99zymPy0dk+gvLkJD+c5GFV9Qv7MeQhsZe1/FSSJ40xnpnkT5N88AHc9qiYso7bn8A+57yl19I+536mPC7Xdr+z9BuFkGRvb7d9Isl7qipJjiV5cVXdPcb4YFU9ONs/qN45xrj+IAZeY0uvZZLnJvm5qnpxkockeWRVvWOMcRSDZMo6/nOSO8YY548qvS9r8oNqJlPW8sFJvjzGuDNJqur6JD+e5B37PfSa2nUte6yNMW6oqj+vqmN7ue0RsvQ6jjHuss+5lymPyefHPqeb+v29nvuduZ8EfZhP2f4HxpeyfZTo/BPTn/H/XP/t+f6LeCrJXye5Zu6vYx1OU9byPtuvyBF+wcTUdUzyD0mevjj/piR/MPfXdBjXMtv/aLs12889rmy/COW1c39N67yWSX4o33/zqsuz/fSUeqB/DxfzaeI62uesaC3vc50jvc9ZxVqu637HEeQJxgXebruqfm3x8Qs+7zjb/wJ9VZLPVtXNi21vHGPcsJ8zr6uJa8nCCtbxtUneWVWXZPsH3i/v68BrbMpajjE+WVXvy/Z/K96d5NM5ZG+zukp7XMuXJ/n1xRH4/03yirG9x9zxtrN8ITObso5V9ROxz/meiY9JmhWs5Vrud7zVNAAANF6kBwAAjUAGAIBGIAMAQCOQAQCgEcgAANAIZAAAaAQyAAA0/wceFVFs3MY9ywAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<Figure size 720x432 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "def normal_random(sample_size=100):\n",
    "    sample = [random.uniform(0,1) for _ in range(sample_size) ]\n",
    "    return sum(sample)/sample_size\n",
    "\n",
    "sample = [normal_random() for _ in range(100)]\n",
    "plt.figure(figsize=(10,6))\n",
    "plt.hist(sample)\n",
    "plt.tight_layout()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Correlation and Evil Baseball Corp\n",
    "\n",
    "Correlation allows us to find relations between data sequences. In our toy example, let's pretend there is an evil baseball corporation that pays its players according to their height - the taller the player is, the more money he/she gets. Suppose there is a base salary of $1000, and an additional bonus from $0 to $100, depending on height. We will take the real players from MLB, and compute their imaginary salaries:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[(74, 1075.2469071629068), (74, 1075.2469071629068), (72, 1053.7477908306478), (72, 1053.7477908306478), (73, 1064.4973489967772), (69, 1021.4991163322591), (69, 1021.4991163322591), (71, 1042.9982326645181), (76, 1096.746023495166), (71, 1042.9982326645181)]\n"
     ]
    }
   ],
   "source": [
    "heights = df['Height']\n",
    "salaries = 1000+(heights-heights.min())/(heights.max()-heights.mean())*100\n",
    "print(list(zip(heights, salaries))[:10])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's now compute covariance and correlation of those sequences. `np.cov` will give us a so-called **covariance matrix**, which is an extension of covariance to multiple variables. The element $M_{ij}$ of the covariance matrix $M$ is a correlation between input variables $X_i$ and $X_j$, and diagonal values $M_{ii}$ is the variance of $X_{i}$. Similarly, `np.corrcoef` will give us the **correlation matrix**."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Covariance matrix:\n",
      "[[  5.31679808  57.15323023]\n",
      " [ 57.15323023 614.37197275]]\n",
      "Covariance = 57.153230230544736\n",
      "Correlation = 1.0\n"
     ]
    }
   ],
   "source": [
    "print(f\"Covariance matrix:\\n{np.cov(heights, salaries)}\")\n",
    "print(f\"Covariance = {np.cov(heights, salaries)[0,1]}\")\n",
    "print(f\"Correlation = {np.corrcoef(heights, salaries)[0,1]}\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "A correlation equal to 1 means that there is a strong **linear relation** between two variables. We can visually see the linear relation by plotting one value against the other:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 720x432 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.figure(figsize=(10,6))\n",
    "plt.scatter(heights,salaries)\n",
    "plt.tight_layout()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's see what happens if the relation is not linear. Suppose that our corporation decided to hide the obvious linear dependency between heights and salaries, and introduced some non-linearity into the formula, such as `sin`:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Correlation = 0.9835304456670837\n"
     ]
    }
   ],
   "source": [
    "salaries = 1000+np.sin((heights-heights.min())/(heights.max()-heights.mean()))*100\n",
    "print(f\"Correlation = {np.corrcoef(heights, salaries)[0,1]}\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In this case, the correlation is slightly smaller, but it is still quite high. Now, to make the relation even less obvious, we might want to add some extra randomness by adding some random variable to the salary. Let's see what happens:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Correlation = 0.9363097848296155\n"
     ]
    }
   ],
   "source": [
    "salaries = 1000+np.sin((heights-heights.min())/(heights.max()-heights.mean()))*100+np.random.random(size=len(heights))*20-10\n",
    "print(f\"Correlation = {np.corrcoef(heights, salaries)[0,1]}\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 720x432 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.figure(figsize=(10,6))\n",
    "plt.scatter(heights, salaries)\n",
    "plt.tight_layout()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "> Can you guess why the dots line up into vertical lines like this?\n",
    "\n",
    "We have observed the correlation between an artificially engineered concept like salary and the observed variable *height*. Let's also see if the two observed variables, such as height and weight, correlate too:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 1., nan],\n",
       "       [nan, nan]])"
      ]
     },
     "execution_count": 26,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.corrcoef(df['Height'],df['Weight'])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Unfortunately, we did not get any results - only some strange `nan` values. This is due to the fact that some of the values in our series are undefined, represented as `nan`, which causes the result of the operation to be undefined as well. By looking at the matrix we can see that `Weight` is the problematic column, because self-correlation between `Height` values has been computed.\n",
    "\n",
    "> This example shows the importance of **data preparation** and **cleaning**. Without proper data we cannot compute anything.\n",
    "\n",
    "Let's use `fillna` method to fill the missing values, and compute the correlation: "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[1.        , 0.52959196],\n",
       "       [0.52959196, 1.        ]])"
      ]
     },
     "execution_count": 27,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.corrcoef(df['Height'],df['Weight'].fillna(method='pad'))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "There is indeed a correlation, but not such a strong one as in our artificial example. Indeed, if we look at the scatter plot of one value against the other, the relation would be much less obvious:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 720x432 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.figure(figsize=(10,6))\n",
    "plt.scatter(df['Height'],df['Weight'])\n",
    "plt.xlabel('Height')\n",
    "plt.ylabel('Weight')\n",
    "plt.tight_layout()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Conclusion\n",
    "\n",
    "In this notebook we have learnt how to perform basic operations on data to compute statistical functions. We now know how to use a sound apparatus of math and statistics in order to prove some hypotheses, and how to compute confidence intervals for arbitrary variables given a data sample. "
   ]
  }
 ],
 "metadata": {
  "interpreter": {
   "hash": "86193a1ab0ba47eac1c69c1756090baa3b420b3eea7d4aafab8b85f8b312f0c5"
  },
  "kernelspec": {
   "display_name": "Python 3 (ipykernel)",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.8.12"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 4
}
